Question: Simplify and expand the following expression: $ \dfrac{3}{x + 1}+ \dfrac{5}{x - 9}- \dfrac{5x}{x^2 - 8x - 9} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{5x}{x^2 - 8x - 9} = \dfrac{5x}{(x + 1)(x - 9)}$ Now we have: $ \dfrac{3}{x + 1}+ \dfrac{5}{x - 9}- \dfrac{5x}{(x + 1)(x - 9)} $ The least common multiple of the denominators is: $ (x + 1)(x - 9)$ In order to get the first term over $(x + 1)(x - 9)$ , multiply by $\dfrac{x - 9}{x - 9}$ $ \dfrac{3}{x + 1} \times \dfrac{x - 9}{x - 9} = \dfrac{3(x - 9)}{(x + 1)(x - 9)} $ In order to get the second term over $(x + 1)(x - 9)$ , multiply by $\dfrac{x + 1}{x + 1}$ $ \dfrac{5}{x - 9} \times \dfrac{x + 1}{x + 1} = \dfrac{5(x + 1)}{(x + 1)(x - 9)} $ Now we have: $ \dfrac{3(x - 9)}{(x + 1)(x - 9)} + \dfrac{5(x + 1)}{(x + 1)(x - 9)} - \dfrac{5x}{(x + 1)(x - 9)} $ $ = \dfrac{ 3(x - 9) + 5(x + 1) - 5x} {(x + 1)(x - 9)} $ Expand: $ = \dfrac{3x - 27 + 5x + 5 - 5x}{x^2 - 8x - 9} $ $ = \dfrac{3x - 22}{x^2 - 8x - 9}$